The n-Vortex Problem for Co-Circular Central Configurations

The Newtonian n-body problem is a well-studied question in celestial mechanics with many practical applications. Of particular interest is the study of central configurations in which each mass accelerates towards the center of mass at a rate proportional to its displacement from the center of mass and where the proportion of the displacement is the same for each mass. Central configurations can be used to find periodic solutions to the n-body problem. One interesting variant of the n-body problem is the study of the central configurations of Helmholtz vortices, whirlpools on an infinite horizontal surface consisting of a perfect fluid.

A configuration in both the n-body and n-vortex problem is central if and only if it satisfies the Albouy-Chenciner (AC) equations.  We use the AC equations to study the central configurations of four vortices from the point of view of algebraic geometry. In a recent paper Cors and Roberts completely characterize the central configurations of four bodies when all four lie on the same circle. We analyze the extent to which these breakthroughs extend to the four vortex case. We also examine some special cases of the four-vortex problem not covered by some recent results of Hampton and Moeckel.